Abstract

We demonstrate that the fractional-order Fourier transformation is a suitable method to analyze the diffraction patterns of particle field holograms. This method permits reconstruction of in-line digital holograms beyond the Fraunhofer condition (d2/λz10). We show that the diameter of spherical particles is measured with good accuracy. Simulation and experimental results are presented.

© 2002 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  3. D. Lebrun, S. Belaı̈d, C. Özkul, “Hologram reconstruction by use of optical wavelet transform,” Appl. Opt. 38, 3730–3734 (1999).
    [CrossRef]
  4. C. Buraga, S. Coëtmellec, D. Lebrun, C. Özkul, “Application of wavelet transform to hologram analysis: three-dimensional location of particles,” Opt. Lasers Eng. 33, 409–421 (2000).
    [CrossRef]
  5. L. Onural, M. T. Özgen, “Extraction of the three-dimensional object location information directly from the in-line holograms using Wigner analysis,” J. Opt. Soc. Am. A 9, 252–260 (1992).
    [CrossRef]
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    [CrossRef] [PubMed]
  7. V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
    [CrossRef]
  8. A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
    [CrossRef]
  9. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation. I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [CrossRef]
  10. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementation. II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
    [CrossRef]
  11. P. Pellat-Finet, “Fresnel diffraction and the fractional-order Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).
    [CrossRef] [PubMed]
  12. P. Pellat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
    [CrossRef]
  13. P. Pellat-Finet, “Transfert du champ électromagnétique par diffraction et transformation de Fourier fractionnaire,” C. R. Acad. Paris, t.320, Série IIb, 91–97 (1995).
  14. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  15. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).
  16. H. M. Nuzzensveig, “High frequency scattering by an inpenetrable sphere,” Ann. Phys. (Leipzig) 33 (1965).
  17. G. A. Tayler, B. J. Thompson, “Fraunhofer holography applied to particle size analysis: a reassessment,” Opt. Acta 23, 261–304 (1976).
  18. A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementation of the two-dimensional fractional Fourier transform with different orders in the two dimensions,” Opt. Commun. 120, 134–138 (1995).
    [CrossRef]
  19. A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementations of two-dimensional fractional Fourier transforms and linear canonical transforms with arbitrary parameters,” Appl. Opt. 37, 2130–2141 (1998).
    [CrossRef]
  20. F. J. Marinho, L. M. Bernardo, “Numerical calculation of fractional Fourier transforms with a single fast-Fourier-transform algorithm,” J. Opt. Soc. Am. A 15, 2111–2116 (1998).
    [CrossRef]
  21. A. W. Lohmann, Bernard H. Soffer, “Relationships between the Radon–Wigner and the fractional Fourier transform,” J. Opt. Soc. Am. A 11, 1798–1801 (1994).
    [CrossRef]
  22. T. M. Kreis, W. P. O. Jüptner, “Suppression of the dc term in digital holography,” Opt. Eng. 36, 2357–2360 (1997).
    [CrossRef]
  23. D. Mendlovic, Z. Zalevsky, R. Dorsch, Y. Bitran, A. Lohmann, H. Ozaktas, “New signal representation based on the fractional Fourier transform: definitions,” J. Opt. Soc. Am. A 12, 2424–2431 (1995).
    [CrossRef]
  24. W. Mecklenbräuker, F. Hlawatsch, The Wigner Distribution. Theory and Applications in Signal Processing (Elsevier, Amsterdam, 1997), pp. 59–83.
  25. R. Bexon, G. D. Bishop, J. Gibbs, “Automatic assessment of aerosol holograms,” J. Aerosol Sci. 7, 397–407 (1976).
    [CrossRef]

2002

2000

C. Buraga, S. Coëtmellec, D. Lebrun, C. Özkul, “Application of wavelet transform to hologram analysis: three-dimensional location of particles,” Opt. Lasers Eng. 33, 409–421 (2000).
[CrossRef]

1999

1998

1997

T. M. Kreis, W. P. O. Jüptner, “Suppression of the dc term in digital holography,” Opt. Eng. 36, 2357–2360 (1997).
[CrossRef]

S. Belaı̈d, D. Lebrun, C. Özkul, “Application of two-dimensional wavelet transform to hologram analysis: visualization of glass fibers in turbulent flame,” Opt. Eng. 36, 1947–1951 (1997).
[CrossRef]

1995

A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementation of the two-dimensional fractional Fourier transform with different orders in the two dimensions,” Opt. Commun. 120, 134–138 (1995).
[CrossRef]

D. Mendlovic, Z. Zalevsky, R. Dorsch, Y. Bitran, A. Lohmann, H. Ozaktas, “New signal representation based on the fractional Fourier transform: definitions,” J. Opt. Soc. Am. A 12, 2424–2431 (1995).
[CrossRef]

1994

1993

1992

1987

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

1980

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

1976

G. A. Tayler, B. J. Thompson, “Fraunhofer holography applied to particle size analysis: a reassessment,” Opt. Acta 23, 261–304 (1976).

R. Bexon, G. D. Bishop, J. Gibbs, “Automatic assessment of aerosol holograms,” J. Aerosol Sci. 7, 397–407 (1976).
[CrossRef]

1965

H. M. Nuzzensveig, “High frequency scattering by an inpenetrable sphere,” Ann. Phys. (Leipzig) 33 (1965).

Belai¨d, S.

D. Lebrun, S. Belaı̈d, C. Özkul, “Hologram reconstruction by use of optical wavelet transform,” Appl. Opt. 38, 3730–3734 (1999).
[CrossRef]

S. Belaı̈d, D. Lebrun, C. Özkul, “Application of two-dimensional wavelet transform to hologram analysis: visualization of glass fibers in turbulent flame,” Opt. Eng. 36, 1947–1951 (1997).
[CrossRef]

Bernardo, L. M.

Bexon, R.

R. Bexon, G. D. Bishop, J. Gibbs, “Automatic assessment of aerosol holograms,” J. Aerosol Sci. 7, 397–407 (1976).
[CrossRef]

Bishop, G. D.

R. Bexon, G. D. Bishop, J. Gibbs, “Automatic assessment of aerosol holograms,” J. Aerosol Sci. 7, 397–407 (1976).
[CrossRef]

Bitran, Y.

Bonnet, G.

P. Pellat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
[CrossRef]

Buraga, C.

C. Buraga, S. Coëtmellec, D. Lebrun, C. Özkul, “Application of wavelet transform to hologram analysis: three-dimensional location of particles,” Opt. Lasers Eng. 33, 409–421 (2000).
[CrossRef]

Coëtmellec, S.

S. Coëtmellec, D. Lebrun, C. Özkul, “Characterization of diffraction patterns directly from in-line holograms with the fractional Fourier transform,” Appl. Opt. 41, 312–319 (2002).
[CrossRef] [PubMed]

C. Buraga, S. Coëtmellec, D. Lebrun, C. Özkul, “Application of wavelet transform to hologram analysis: three-dimensional location of particles,” Opt. Lasers Eng. 33, 409–421 (2000).
[CrossRef]

Dorsch, R.

Gibbs, J.

R. Bexon, G. D. Bishop, J. Gibbs, “Automatic assessment of aerosol holograms,” J. Aerosol Sci. 7, 397–407 (1976).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Hlawatsch, F.

W. Mecklenbräuker, F. Hlawatsch, The Wigner Distribution. Theory and Applications in Signal Processing (Elsevier, Amsterdam, 1997), pp. 59–83.

Jüptner, W. P. O.

T. M. Kreis, W. P. O. Jüptner, “Suppression of the dc term in digital holography,” Opt. Eng. 36, 2357–2360 (1997).
[CrossRef]

Kerr, F. H.

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

Kreis, T. M.

T. M. Kreis, W. P. O. Jüptner, “Suppression of the dc term in digital holography,” Opt. Eng. 36, 2357–2360 (1997).
[CrossRef]

Lebrun, D.

S. Coëtmellec, D. Lebrun, C. Özkul, “Characterization of diffraction patterns directly from in-line holograms with the fractional Fourier transform,” Appl. Opt. 41, 312–319 (2002).
[CrossRef] [PubMed]

C. Buraga, S. Coëtmellec, D. Lebrun, C. Özkul, “Application of wavelet transform to hologram analysis: three-dimensional location of particles,” Opt. Lasers Eng. 33, 409–421 (2000).
[CrossRef]

D. Lebrun, S. Belaı̈d, C. Özkul, “Hologram reconstruction by use of optical wavelet transform,” Appl. Opt. 38, 3730–3734 (1999).
[CrossRef]

S. Belaı̈d, D. Lebrun, C. Özkul, “Application of two-dimensional wavelet transform to hologram analysis: visualization of glass fibers in turbulent flame,” Opt. Eng. 36, 1947–1951 (1997).
[CrossRef]

Lohmann, A.

Lohmann, A. W.

Marinho, F. J.

McBride, A. C.

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

Mecklenbräuker, W.

W. Mecklenbräuker, F. Hlawatsch, The Wigner Distribution. Theory and Applications in Signal Processing (Elsevier, Amsterdam, 1997), pp. 59–83.

Mendlovic, D.

Namias, V.

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

Nuzzensveig, H. M.

H. M. Nuzzensveig, “High frequency scattering by an inpenetrable sphere,” Ann. Phys. (Leipzig) 33 (1965).

Onural, L.

Ozaktas, H.

Ozaktas, H. M.

Özgen, M. T.

Özkul, C.

S. Coëtmellec, D. Lebrun, C. Özkul, “Characterization of diffraction patterns directly from in-line holograms with the fractional Fourier transform,” Appl. Opt. 41, 312–319 (2002).
[CrossRef] [PubMed]

C. Buraga, S. Coëtmellec, D. Lebrun, C. Özkul, “Application of wavelet transform to hologram analysis: three-dimensional location of particles,” Opt. Lasers Eng. 33, 409–421 (2000).
[CrossRef]

D. Lebrun, S. Belaı̈d, C. Özkul, “Hologram reconstruction by use of optical wavelet transform,” Appl. Opt. 38, 3730–3734 (1999).
[CrossRef]

S. Belaı̈d, D. Lebrun, C. Özkul, “Application of two-dimensional wavelet transform to hologram analysis: visualization of glass fibers in turbulent flame,” Opt. Eng. 36, 1947–1951 (1997).
[CrossRef]

Pellat-Finet, P.

P. Pellat-Finet, “Fresnel diffraction and the fractional-order Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).
[CrossRef] [PubMed]

P. Pellat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
[CrossRef]

P. Pellat-Finet, “Transfert du champ électromagnétique par diffraction et transformation de Fourier fractionnaire,” C. R. Acad. Paris, t.320, Série IIb, 91–97 (1995).

Sahin, A.

A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementations of two-dimensional fractional Fourier transforms and linear canonical transforms with arbitrary parameters,” Appl. Opt. 37, 2130–2141 (1998).
[CrossRef]

A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementation of the two-dimensional fractional Fourier transform with different orders in the two dimensions,” Opt. Commun. 120, 134–138 (1995).
[CrossRef]

Soffer, Bernard H.

Tayler, G. A.

G. A. Tayler, B. J. Thompson, “Fraunhofer holography applied to particle size analysis: a reassessment,” Opt. Acta 23, 261–304 (1976).

Thompson, B. J.

G. A. Tayler, B. J. Thompson, “Fraunhofer holography applied to particle size analysis: a reassessment,” Opt. Acta 23, 261–304 (1976).

Zalevsky, Z.

Ann. Phys. (Leipzig)

H. M. Nuzzensveig, “High frequency scattering by an inpenetrable sphere,” Ann. Phys. (Leipzig) 33 (1965).

Appl. Opt.

IMA J. Appl. Math.

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

J. Aerosol Sci.

R. Bexon, G. D. Bishop, J. Gibbs, “Automatic assessment of aerosol holograms,” J. Aerosol Sci. 7, 397–407 (1976).
[CrossRef]

J. Inst. Math. Appl.

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Acta

G. A. Tayler, B. J. Thompson, “Fraunhofer holography applied to particle size analysis: a reassessment,” Opt. Acta 23, 261–304 (1976).

Opt. Commun.

A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementation of the two-dimensional fractional Fourier transform with different orders in the two dimensions,” Opt. Commun. 120, 134–138 (1995).
[CrossRef]

P. Pellat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
[CrossRef]

Opt. Eng.

S. Belaı̈d, D. Lebrun, C. Özkul, “Application of two-dimensional wavelet transform to hologram analysis: visualization of glass fibers in turbulent flame,” Opt. Eng. 36, 1947–1951 (1997).
[CrossRef]

T. M. Kreis, W. P. O. Jüptner, “Suppression of the dc term in digital holography,” Opt. Eng. 36, 2357–2360 (1997).
[CrossRef]

Opt. Lasers Eng.

C. Buraga, S. Coëtmellec, D. Lebrun, C. Özkul, “Application of wavelet transform to hologram analysis: three-dimensional location of particles,” Opt. Lasers Eng. 33, 409–421 (2000).
[CrossRef]

Opt. Lett.

Other

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

P. Pellat-Finet, “Transfert du champ électromagnétique par diffraction et transformation de Fourier fractionnaire,” C. R. Acad. Paris, t.320, Série IIb, 91–97 (1995).

W. Mecklenbräuker, F. Hlawatsch, The Wigner Distribution. Theory and Applications in Signal Processing (Elsevier, Amsterdam, 1997), pp. 59–83.

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Figures (8)

Fig. 1
Fig. 1

Representation of the system axes.

Fig. 2
Fig. 2

Setup for performing the FRFT according to Lohmann’s definition.

Fig. 3
Fig. 3

Optimal value of a versus z.

Fig. 4
Fig. 4

(a) Intensity distribution of the diffraction pattern of a particle, d = 100   μ m and z = 100   mm centered at τ x = τ y = 1   mm . (b) FRFT of the intensity distribution in a opt = 0.5 fractional domain located at τ x a opt = τ y a opt = 0.707   mm .

Fig. 5
Fig. 5

(a) Estimated diameters versus theoretical diameters. (b) Dispersion of estimated diameters versus theoretical diameters.

Fig. 6
Fig. 6

(a) Intensity distribution I ( x ,   y ) of a diffraction pattern produced by a plane seeded with a polysized particle field. (b) Digital reconstruction computed from I ( x ,   y ) . a opt = 0.5 . (c) Area details of Fig. 5(b).

Fig. 7
Fig. 7

(a) Experimental diffraction pattern of a particle diameter d = 600   μ m . z = 102.3   mm , N x = N y = 522 . (b) Reconstruction of the experimental intensity distribution. a opt = 0.507 .

Fig. 8
Fig. 8

Estimated diameter versus theoretical diameter for different axial distances.

Equations (41)

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A ( x ,   y ) = exp i 2 π λ z i λ z - + - + [ 1 - t ( ξ ,   η ) ] × exp i π λ z [ ( ξ - x ) 2 + ( η - y ) 2 ] d ξ d η ,
t ( ξ ,   η ) = 1 if   ξ 2 + η 2 < d / 2 1 / 2 if   ξ 2 + η 2 = d / 2 0 otherwise .
A ( ρ ,   ϕ ) = exp i 2 π λ z i λ z   exp i π λ z ρ 2 d / 2 +   exp i π λ z r 2 × 0 2 π   exp - i 2 π λ z ρ r   cos ( θ - ϕ ) d θ r d r .
0 2 π   exp - i 2 π λ z ρ r   cos ( θ - ϕ ) d θ = 2 π J 0 2 π ρ r λ z .
A ( ρ ) = 2 π exp i 2 π λ z i λ z   exp i π λ z ρ 2 × d / 2 + J 0 2 π ρ r λ z exp i π λ z r 2 r d r .
1 + J 0 ( vt ) exp iu 2 t 2 t d t = i u   exp iu 2 [ V 0 ( u ,   v ) + iV 1 ( u ,   v ) ] ,
V k ( u ,   v ) = cos u 2 + v 2 2 u + k π 2 + m = 0 ( - 1 ) m u v 2 m - k + 2 J 2 m - k + 2 ( v ) ,
lim v 0 J n ( α v ) ( α v ) n = 1 2 n n ! .
V 0 ( u ,   v ) = cos u 2 + v 2 2 u + m = 0 ( - 1 ) m u v 2 m + 2 J 2 m + 2 ( v ) ,
V 1 ( u ,   v ) = cos u 2 + v 2 2 u + π 2 + m = 0 ( - 1 ) m u v 2 m + 1 J 2 m + 1 ( v ) .
A ( ρ ) = exp i 2 π λ z exp i π λ z ( ρ 2 + ( d / 2 ) 2 × V 0 π d 2 2 λ z ,   π d ρ λ z + iV 1 π d 2 2 λ z ,   π d ρ λ z .
I ( ρ ) = V 0 2 π d 2 2 λ z ,   π d ρ λ z + V 1 2 π d 2 2 λ z ,   π d ρ λ z .
I ( x ,   y ) = 1 - 2 λ z   sin π λ z ( x 2 + y 2 ) F ( x ,   y ) + 1 ( λ z ) 2 F 2 ( x ,   y ) ,
F ( x ,   y ) = π d 2 2 J 1 [ ( π d x 2 + y 2 ) / λ z ] ( π d x 2 + y 2 ) / λ z .
F α 1 , α 2 [ f ( x ,   y ) ] ( x a 1 ,   y a 2 ) = - + - + K α 1 , α 2 ( x ,   y ;   x a 1 ,   y a 2 ) f ( x ,   y ) d x d y ,
α 1 = a 1 π 2 , α 2 = a 2 π 2 ,
K α 1 , α 2 ( x ,   y ;   x a 1 ,   y a 2 ) = K α 1 ( x ,   x a 1 ) K α 2 ( y ,   y a 2 )
K α 1 ( x ,   x a 1 ) = C ( α 1 ) exp i π cot   α 1 λ f 1 ( x 2 + x a 1 2 ) × exp - i 2 π xx a 1 λ f 1   sin   α 1
K α 2 ( y ,   y a 2 ) = C ( α 2 ) exp i π cot   α 2 λ f 1 ( y 2 + y a 2 2 ) × exp - i 2 π yy a 2 λ f 1   sin   α 2 .
C ( α 1 ) = exp { - i [ ( π / 4 )   sign ( sin   α 1 ) - α 1 / 2 ] } | λ f 1   sin   α 1 | 1 / 2 ,
C ( α 2 ) = exp { - i [ ( π / 4 )   sign ( sin   α 2 ) - α 2 / 2 ] } | λ f 1   sin   α 2 | 1 / 2 .
F α 1 , α 2 [ f ( x - τ x ,   y - τ y ) ] ( x a 1 ,   y a 2 ) = exp - i 2 π τ x   sin   α 1 ( x a 1 - τ x / 2   cos   α 1 ) λ f 1 × exp - i 2 π τ y   sin   α 2 ( y a 2 - τ y / 2   cos   α 2 ) λ f 1 × F α 1 , α 2 [ f ( x ,   y ) ] ( x a 1 - τ x   cos   α 1 ,   y a 2 - τ y   cos   α 2 ) .
W f a 1 , a 2 ( x a 1 ,   y a 2 ; u a 1 ,   v a 2 ) = R - α 1 , - α 2 { W f ( x ,   y ; u ,   v ) } ,
R α 1 { R α 2 [ W f ( x ,   y ,   u ,   v ) ] } ( x a 1 ,   y a 2 ) = | F α 1 , α 2 [ f ( x ,   y ) ] ( x a 1 ,   y a 2 ) | 2 .
- 2 λ z   sin π λ z ( x 2 + y 2 ) F ( x ,   y ) = f ( x ,   y ) - g ( x ,   y ) ,
f ( x ,   y ) = F ( x ,   y ) i λ z   exp - i π λ z ( x 2 + y 2 )
g ( x ,   y ) = F ( x ,   y ) i λ z   exp i π λ z ( x 2 + y 2 ) .
h ( x ,   y ) = 1 ( λ z ) 2 F 2 ( x ,   y ) .
F α 1 , α 2 [ I c ( x ,   y ) ] ( x a 1 ,   y a 2 ) = f a 1 , a 2 ( x a 1 ,   y a 2 ) - g a 1 , a 2 ( x a 1 ,   y a 2 ) + h a 1 , a 2 ( x a 1 ,   y a 2 ) .
β 1 = cot   α 1 - f 1 / z = cot   θ 1 ,
β 2 = cot   α 2 - f 1 / z = cot   θ 2 ,
1 sin   θ 1 = ( β 1 2 + 1 ) 1 / 2 , 1 sin   θ 2 = ( β 2 2 + 1 ) 1 / 2 .
f a 1 , a 2 ( x a 1 ,   y a 2 ) = C ( α 1 ) C ( α 2 ) i λ z   exp i π λ f 1 ( x a 1 2   cot   α 1 + y a 2 2   cot   α 2 ) × - + - + F ( x ,   y ) exp i π λ f 1 ( x 2   cot   θ 1 + y 2   cot   θ 2 ) × exp - i 2 π λ f 1 s 1 x a 1 x sin   θ 1 + s 2 y a 2 y sin   θ 2 d x d y .
f ˆ a 1 , a 2 ( x a 1 ,   y a 2 ) = φ ( x a 1 ,   y a 2 ) F θ 1 , θ 2 [ F ( x ,   y ) ] ( x a 1 ,   y a 2 ) ,
f ˆ a 1 , a 2 ( x a 1 ,   y a 2 ) = f a 1 , a 2 ( x a 1 / s 1 ,   y a 2 / s 2 )
g ˆ a 1 , a 2 ( x a 1 ,   y a 2 ) = φ ( x a 1 ,   y a 2 ) F θ 1 , θ 2 F ( x ,   y ) × exp i 2 π λ z ( x 2 + y 2 ) ( x a 1 ,   y a 2 ) .
h ˆ a 1 , a 2 ( x a 1 ,   y a 2 ) = i φ ( x a 1 ,   y a 2 ) λ z F θ 1 , θ 2 F 2 ( x ,   y ) × exp i π λ z ( x 2 + y 2 ) ( x a 1 ,   y a 2 ) .
F ˆ α 1 , α 2 [ I c ( x ,   y ) ] ( x a 1 ,   y a 2 ) = f ˆ a 1 , a 2 ( x a 1 ,   y a 2 ) - g ˆ a 1 , a 2 ( x a 1 ,   y a 2 ) + h ˆ a 1 , a 2 ( x a 1 ,   y a 2 ) ,
F ˆ α 1 , α 2 [ I c ( x ,   y ) ] ( x a 1 ,   y a 2 ) = F α 1 , α 2 [ I c ( x ,   y ) ] ( x a 1 / s 1 ,   y a 2 / s 2 ) .
cot   α opt = cot   α 1 = cot   α 2 = f 1 / z .
| F ˆ α 1 , α 2 [ I c ( x ,   y ) ] ( x a 1 ,   y a 2 ) | 2 = | f ˆ a 1 , a 2 ( x a 1 ,   y a 2 ) | 2 + | g ˆ a 1 , a 2 ( x a 1 ,   y a 2 ) | 2 + | h ˆ a 1 ,   a 2 ( x a 1 ,   y a 2 ) | 2 + I .

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